Question

The value of ${\int }_0^{10}{e}^{\mathrm{sgn}(x-[x\left]\right)}dx$ equals (where $[.]$ represents the greatest integer function)

• A. $e^{10}$
• B. $e^{-10}$
• C. $-10e$
• D. $10e$

Answer 1

Answer: (D)

$10e$

${\int }_0^{10}{e}^{sgn(x-[x\left]\right)}dx$

$={\int }_0^{10}{e}^{sgn\left\{x\right\}}dx$ $sgn\left(x\right)=\{\begin{array}{cc}1,& x>0\\ 0,& x=0\\ -1,& x<0\end{array}$

$={\int }_0^{10}{e}^{\prime }dx$ $(\because \{x\}>0)$

$=ex{|}_0^{10}$

$=10e$